WASHINGTON — President Obama and House Republicans failed to reach agreement on a six-week extension of the nation’s borrowing authority during a meeting Thursday at the White House, but the two sides kept talking, and the offer from politically besieged Republicans was seen as an initial step toward ending the budget standoff.

In statements afterward that struck the most positive tone in weeks of acrimony, House Republicans described their hour-and-a-half-long meeting with Mr. Obama as “a useful and productive conversation,” while the White House described “a good meeting,” though “no specific determination was made” about the Republicans’ offer. Both agreed to continue talks through the night.

People familiar with the meeting said that Mr. Obama pressed Republicans to reopen the government, and that Republicans raised the possibility that financing could be restored by early next week if terms for broad budget negotiations could be reached.

Twenty Republicans, led by Speaker John A. Boehner, went to the White House at Mr. Obama’s invitation after a day of fine-tuning their proposal to increase the Treasury Department’s authority to borrow money to pay existing obligations through Nov. 22. The government is expected to reach its borrowing limit next week. In exchange, they sought a commitment by the president to negotiate a deal for long-term deficit reduction and a tax overhaul.

The president “didn’t say yes, didn’t say no,” said Representative Paul D. Ryan, Republican of Wisconsin and chairman of the House Budget Committee. He added, “We agreed to continue talking and continue negotiating.”

Still, the House Republican offer represented a potentially significant breakthrough. Even if Democrats found fault with the Republicans’ immediate proposal — for example, it would prevent the Treasury secretary from engaging in accounting maneuvers to stave off potential default — it was seen as an opening gambit in the legislative dance toward some resolution before the government is expected to breach its debt limit on Thursday.

Even before the meeting, the White House and its Democratic allies in Congress were all but declaring victory at the evidence that Republicans — suffering the most in polls, and pressured by business allies and donors not to provoke a government default — were seeking a way out of the impasse.

After some fretful weeks, the Democrats believe, Mr. Obama was seeing some payoff for his big gamble this year. Burned by his experience with House Republicans in mid-2011, when brinkmanship over the debt limit hobbled the already weak economy, Mr. Obama began his second term vowing never again to negotiate over raising the ceiling or to give any concessions to Republicans for performing an act that is their constitutional responsibility.

“The good news is that Republicans have accepted the principle that they’re not going to attach conditions to the debt ceiling,” said Representative Chris Van Hollen of Maryland, the senior Democrat on the House Budget Committee. “The bad news is they’ve only extended the debt ceiling for six weeks.”

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Speaker John A. Boehner departed the Capitol on Thursday for a meeting at the White House with President Obama about a solution to ending the government shutdown.CreditJ. Scott Applewhite/Associated Press

For House Republicans, the maneuvers represented a near-reversal of their original strategy in September of going to the mat over the debt limit but not shutting down the government. Now, under pressure from falling poll numbers and angry business supporters, they are seeking a compromise on the debt ceiling. Yet for now, they are still refusing to finance and reopen the government without some concessions.

Mr. Boehner and his colleagues left the White House without speaking to waiting reporters, and quickly gathered in his Capitol suite for further discussion. Their debt limit proposal could come to a vote as soon as Friday.

Before the White House meeting, administration and Congressional Democrats said they were skeptical that House Republican leaders could pass the proposal. A large faction of Tea Party conservatives campaigned on promises never to vote to increase the nation’s debt limit. And Congressional Democrats vowed to oppose any proposal that did not also fully finance a government now shuttered since the fiscal year began Oct. 1.

“We’ll see what they’re able to pass,” said Mr. Obama’s press secretary, Jay Carney.

Senate Democrats had their own White House meeting with Mr. Obama and Vice President Joseph R. Biden Jr. three hours before the House Republicans arrived, and the majority leader, Senator Harry Reid of Nevada, declined to embrace the Republicans’ debt limit proposal until he saw it. He told reporters that Democrats would not negotiate on further deficit reductions until House Republicans agreed to the measure passed by the Senate to finance and open the government through mid-November.

“Not going to happen,” Mr. Reid said. “Open the government,” he added. “There is so much pain and suffering out there. It is really tear-jerking, to say the least.”

Separately, members of the Senate’s Republican minority, who are to meet with Mr. Obama on Friday, worked on a proposal for full-year government funding — at levels reflecting the across-the-board spending cuts known as sequestration, but giving agencies flexibility to shift money around. They are considering adding it to any short-term debt limit increase that the House might pass and send to the Senate.

Mr. Ryan said before the White House meeting that Republicans were now willing to formally negotiate with Senate Democrats over a long-term, comprehensive budget framework. The Republicans have resisted such a move since April, fearing that it would require compromises, like raising additional tax revenues, that would enrage the party’s conservative base heading into the 2014 midterm elections.

Many House Republicans, leaving a closed-door party caucus earlier Thursday that at times grew contentious, said they would support their leadership’s short-term debt limit proposal. But they said they would do so only if Mr. Obama agreed to negotiate a broader deficit reduction deal, with big savings from entitlement programs.

The president has insisted he will not agree to significant reductions in projected Medicare and Medicaid spending — even his own tentative proposals — unless Republicans agree to raise revenues by curbing tax breaks for corporations and wealthy individuals. And Mr. Boehner in recent days reaffirmed the party’s anti-tax stance, which suggests that future talks could founder.

Economists across a broad spectrum agree that breaching the debt limit would damage the economy. The new Republican proposal could temporarily remove that threat.

Jeremy W. Peters and Annie Lowrey contributed reporting.

A version of this article appears in print on , on Page A1 of the New York edition with the headline: Talks Are Begun On Fiscal Crisis; No Quick Accord. Order Reprints | Today’s Paper | Subscribe